University of California Los Angeles The Critical Casimir Effect in Model Physical Systems A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Physics by Jonathan Ariel Bergknoff 2012 ⃝c Copyright by Jonathan Ariel Bergknoff 2012 Abstract of the Dissertation The Critical Casimir Effect in Model Physical Systems by Jonathan Ariel Bergknoff Doctor of Philosophy in Physics University of California, Los Angeles, 2012 Professor Joseph Rudnick, Chair The Casimir effect is an interaction between the boundaries of a finite system when fluctua- tions in that system correlate on length scales comparable to the system size. In particular, the critical Casimir effect is that which arises from the long-ranged thermal fluctuation of the order parameter in a system near criticality. Recent experiments on the Casimir force in binary liquids near critical points and 4He near the superfluid transition have redoubled theoretical interest in the topic. It is an unfortunate fact that exact models of the experi- mental systems are mathematically intractable in general. However, there is often insight to be gained by studying approximations and toy models, or doing numerical computations. In this work, we present a brief motivation and overview of the field, followed by explications of the O(2) model with twisted boundary conditions and the O(n ! 1) model with free boundary conditions. New results, both analytical and numerical, are presented. ii The dissertation of Jonathan Ariel Bergknoff is approved. Giovanni Zocchi Alex Levine Lincoln Chayes Joseph Rudnick, Committee Chair University of California, Los Angeles 2012 iii To my parents, Hugh and Esther Bergknoff iv Table of Contents 1 Introduction :::::::::::::::::::::::::::::::::::::: 1 1.1 The Casimir Effect . 1 1.2 Casimir's Seminal Computation . 3 1.3 An Even Simpler System Exhibiting The Casimir Effect . 5 1.4 The Critical Casimir Effect . 9 1.5 Finite-Size Scaling Theory . 11 1.6 The Casimir Force in Critical Films . 13 1.7 Results on Critical Casimir Forces . 14 1.8 Experimental Verification of the Casimir Effect . 17 2 The Casimir Force in the XY Model with Twisted Boundary Conditions 20 2.1 Introduction . 20 2.2 The Casimir Force in the Three-Dimensional Mean-Field XY Model on a Lattice 23 2.3 The Casimir Force in the Ginzburg-Landau Mean-Field Theory of the Three- Dimensional XY Model . 26 2.4 Numerical Solution of the Model . 35 2.4.1 Numerical Solution of the Lattice Model . 35 2.4.2 Numerical Solution of the Continuum Model . 38 2.5 The Transition at = ............................. 41 2.6 Discussion and Concluding Remarks . 45 3 The Casimir Force in the O(n) Model with n ! 1 :::::::::::::: 47 3.1 Introduction . 47 v 3.2 The Model . 51 3.3 The Bulk System . 55 3.4 The Finite-Size System . 57 3.5 Numerical Solution of the Model . 59 3.6 Discussion and Concluding Remarks . 62 A Free Energy in the Lattice XY Mean-Field Model :::::::::::::: 66 B Derivatives of the Ginzburg-Landau Free Energy Functional ::::::: 68 C Newton's Method :::::::::::::::::::::::::::::::::: 72 D Analytical Results for the XY Ginzburg-Landau Model Under Twisted Boundary Conditions :::::::::::::::::::::::::::::::::: 74 E Low-Temperature Asymptotics of the XY Ginzburg-Landau Model Under Twisted Boundary Conditions :::::::::::::::::::::::::::: 81 F Determining the Kink Temperature ::::::::::::::::::::::: 83 G Some Integrals of Interest ::::::::::::::::::::::::::::: 86 H Spectral Derivatives ::::::::::::::::::::::::::::::::: 89 I Asymptotic Solution of the O(n) Model with n ! 1 ::::::::::::: 91 I.1 Setting Up the Asymptotic Approach . 91 I.2 Casimir Force Considerations . 95 I.3 Zero-Temperature Casimir Force . 97 I.4 T > 0 Correction to Casimir Force . 101 vi I.4.1 The Quantity B1 ............................. 102 I.4.2 Sum of Eigenvalues to Second Order . 104 I.4.3 Putting Together the T > 0 Correction . 106 References ::::::::::::::::::::::::::::::::::::::::: 108 vii List of Figures 1.1 Schematic phase diagram of a binary mixture. 10 2.1 Renderings of the moments (d = 3, N = 30, = ) for temperatures above (left) and below (right) the temperature where a kink occurs in the Casimir force. 24 2.2 The Casimir force in the lattice model (d = 3, N = 30) as a function of reduced temperature t for various values of the twist angle, . 25 2.3 A plot of the dimensionless scaling function for the Casimir force, XCas, versus xt (also dimensionless) for several values of . The dotted curve is the Casimir scaling function in the Ising-like case of a critical fluid under (+; −) boundary conditions. For xt above a certain value, this curve coincides with the one for the model studied here when twisted by an angle ≈ . 31 2.4 The scaling functions for the Casimir force in the Ginzburg-Landau model (solid curves), overlaid with those from the lattice model with N = 50 (brown points), for several values of (from top to bottom: = 0:98, = 2=3, = =2, = =3, = 0). 33 2.5 Casimir force curves in the Ginzburg-Landau model for several values of , overlaid with their respective asymptotic expressions (dotted) given by Eqn. (2.33) as proven in appendix C. 34 ( ) 2.6 Left: The positions xt;0 of the zeros of the Casimir force in the (xt; )-plane. ( ) Right: The Casimir amplitude ΔCas as a function of the twist angle . This curve was initially reported by Krech[24], and follows from the results of ap- pendix D. The Casimir amplitude changes sign at = =3. 35 viii 2.7 Graphical solution of Eqn. (2.52) with = 3=4 and several values of . As is decreased, the number of solutions changes. Thick curve ( = −1): > −2 always has one solution. Dashed curve ( = −2:2): min < < −2 has two solutions, one at p ≈ 0:5 and the other at p ≈ 4. Dotted curve ( ≈ −2:915): = min has only one solution. Thin, solid curve ( = −3:5): < min has no solutions. 40 2.8 The dimensionless quantities X0 (solid), X'=10 (long dashing; scale reduced for ease of plotting) and X'=X0 (short dashing) as functions of xt when ≈ . The vertical dotted line indicates xt = xt;kink................... 42 3.1 The Casimir force in the O(n ! 1) model with free boundary conditions in a three-dimensional film geometry, focusing on the critical region x ≈ 0. The curves, from top to bottom, are for system sizes L = 10, L = 20, L = 30, L = 50, L = 100, L = 200, and L = 500. The scaled Casimir pressure, in 3 units of kBT=L , is plotted against the scaling variable x = Lt. 62 3.2 The Casimir force for L = 10: numeric results (solid, blue) compared to the asymptotic expression (red, dashed) given by Eqn. (3.65). The asymptotic result is valid in the vicinity of x = −L. With L = 10, it offers stunning agreement with the numerics even into the critical region just below x = 0. 64 B.1 A schematic illustration of the change suffered by the angle profile '(z) when L is increased by L. The old and new profiles share the same boundary con- dition, but the boundary moves. We are interested in the quantity '(L=2), indicated by ' in the figure, which is the change in '(L=2). 71 D.1 The loci of points in the (p; )-plane for which the roots x± possess the prop- erties discussed in the text. When any of the roots approaches the thick blue 2 line, X0 ! 1. This is only possible for < 0, i.e. when X0 = xt ! −∞. 75 ix D.2 Plots of the amplitude profile and the angle of the order parameter for = =3 and some choices of xt. We observe that, when the temperature increases, the value of the amplitude in the middle of the system decreases. The twist of the local variables through the system spans over the total system almost uniformly for low temperatures, while for higher ones it concentrates more and more in the middle of the system where the amplitude is at its smallest values. 80 x Acknowledgments I would like to thank Joseph Rudnick for his valuable guidance in my research. I learned a great deal from his wealth of experience and insight. I also thank Daniel Dantchev for much the same reasons. I deeply appreciate all the support given to me by parents, Hugh and Esther Bergknoff, and my wonderful fianc´ee,Janine Adelberg. Thank you to my department friends, Jonathan Landy, Koji Sato, Jonghyoun Eun and Xi "Darktree" Cheng, with whom I had countless interesting and enlightening conversations. I also thank Jenny Lee, who was a tremendous help with all administrative issues. Finally, I thank Robert Finkelstein and the estate of Leo Delsasso, for the financial support provided by their fellowships. xi Vita 2006 B.A. (Physics) and B.A. (Mathematics), Cornell University. 2008 M.S. (Physics), UCLA. xii CHAPTER 1 Introduction 1.1 The Casimir Effect In 1948, Casimir and Polder published an article[1] considering the effect of quantum elec- trodynamics on the van der Waals force between a neutral atom and an infinite, perfectly conducting plane. The classical result is that the interaction energy of this system goes as L−3, with L the distance between the atom and the plane. Casimir and Polder recovered this result for small L, but found the energy to go as L−4 at large L. They argued that the decrease was the result of retardation effects (i.e. of the finite speed of light). Just months later, Casimir revisited the problem on his own. He showed that, by con- sidering the zero point energy of the quantized electromagnetic field under simple boundary conditions, the same sort of result could be obtained with less computational difficulty. More- over, this formulation of the problem is relevant in a far broader class of physical systems.